Abstract: Random graphs are at the intersection of probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices. Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices. The general model that describes this framework is called the exponential random graph model (ERGM). It is used in social network analysis and appears in statistical physics as in the ferromagnetic Ising model. It can also be thought of as a generalization of a p-spin infinite-range spin glass model. We characterize the parameters that determine when an ERGM has desirable properties (e.g., stable, Lorentzian) using a well-developed dictionary between probability distributions and their corresponding generating polynomials.